A394120 G.f. A(x) satisfies A(x)^4 + (x-1)*A(x)^3 + 3*x*A(x)^2 + (x^2-x)*A(x) + x^2 = 0.

0, 1, 3, 14, 77, 460, 2892, 18838, 125999, 860410, 5974998, 42074616, 299785646, 2157615484, 15664666260, 114596758656, 843977389437, 6252656070700, 46568577487040, 348479403676142, 2618857896150888, 19756973221959766, 149571500476890750, 1135960161318209766

Offset

0,3

Comments

Number of n-vertex planar rooted trees with vertices colored red, blue, and green with blue root where red vertices can only occur as leaves, blue vertices can be followed by vertices of any color, and green vertices can be followed by red or green vertices.

Links

Nathan Fox, Table of n, a(n) for n = 0..300

S. Dimitrov, N. Fox, K. Hadaway, A. Tharp, and S. Wagner, Counting Colored Trees, arXiv:2602.16055 [math.CO], 2026.

Formula

a(n) ~ 41^(1/4) * 2^(3*n - 5/2) / (sqrt(3*Pi) * n^(3/2) * (33 - 5*sqrt(41))^(n - 1/2)). - Vaclav Kotesovec, Jun 04 2026

Maple

series(RootOf(A^4+(x-1)*A^3+3*x*A^2+(x^2-x)*A+x^2, A), x, 24):
gfun[seriestolist](%)[];  # Alois P. Heinz, Mar 15 2026

Prog

(Python)
def A394120(n):
    A = [[0, 0, 0], [1, 1, 1], [1, 0, 1]]
    if n == 0:
        return 0
    m = len(A)
    output = [[1] for i in range(m)]
    for l in range(2, n + 1):
        for i in range(m):
            term = 0
            for k in range(1, l):
                for j in range(m):
                    term += A[i][j] * output[i][k - 1] * output[j][l - k - 1]
            output[i].append(term)
    return output[1][n - 1]

Crossrefs

Sequence in context: A198656 A240402 A390686 * A048779 A369477 A394126

Adjacent sequences: A394117 A394118 A394119 * A394121 A394122 A394123

Keyword

nonn

Author

Nathan Fox, Mar 11 2026

Extensions

Definition corrected by Georg Fischer, Mar 15 2026

Status

approved